Question

An amount of money \(a\) invested at a compound interest rate of \(n\) per year will take \(t\) years to accumulate to an amount \(y,\) where \(t\) is $$t=\frac{\log y-\log a}{\log (1+n)} \quad \text { (years) }$$ How many years will it take an investment of \(\$ 10,000\) to triple in value when deposited at \(8.00 \%\) per year? (The equation in this problem is derived from the compound interest formula. The equations in problems 50 and 51 are obtained from the equations for an annuity and for capital recovery from Exercise 1.)

Short answer

It will take approximately 14.27 years for an investment of $10,000 to triple in value at an 8% compound interest rate.

Step-by-step solution

Identify the Given Variables

In this problem, we are given the initial investment amount, which is \(a = \)10,000, the target amount which is three times the initial amount, therefore \(y = 3 \times 10,000 = \)30,000, and the annual compound interest rate, which is \(n = 8\% = 0.08\).

Convert the Percentage to a Decimal

The interest rate is given as a percentage. Convert it into decimal form by dividing by 100. We've got \(n = 8\% = \frac{8}{100} = 0.08\).

Plug the Values into the Formula

Substitute \(a\), \(y\), and \(n\) into the given formula to calculate \(t\) years: \(t = \frac{\log y - \log a}{\log (1 + n)}\). Plugging in the values, we get \(t = \frac{\log (30000) - \log (10000)}{\log (1 + 0.08)}\).

Simplify the Logarithm Expressions

Use the logarithm properties to simplify the expressions: \(t = \frac{\log (30000) - \log (10000)}{\log (1.08)}\). Since the logarithms have the same base, use the property \(\log b - \log a = \log(\frac{b}{a})\) to simplify: \(t = \frac{\log (\frac{30000}{10000})}{\log (1.08)}\).

Calculate the Numerator

Simplify the fraction inside the log in the numerator to find the log value: \(\log (\frac{30000}{10000}) = \log (3)\).

Calculate the Years Needed to Triple the Investment

Now, calculate \(t\) using the logarithm values: \(t = \frac{\log (3)}{\log (1.08)}\). Use a calculator to find the precise value.

Calculate the Final Answer

Using a calculator, compute the right side to find the number of years: \(t \approx \frac{\log (3)}{\log (1.08)} \approx \frac{0.4771}{0.0334} \approx 14.27.\) Round the answer to the nearest whole year if necessary.

Key concepts

Logarithmic Equations

Logarithmic equations are essential in understanding how compound interest works. A logarithm answers the question: to what power must a certain base be raised to obtain a given number? For instance, if we ask what power we must raise 10 to get 1000, we write this as \( \log_{10}(1000) \), which is 3 because \( 10^{3} = 1000 \).

When solving financial problems like calculating the time it takes for an investment to reach a certain value, logarithms help us reverse the exponentiation process found in the compound interest formula. Applying logarithmic properties simplifies equations and enables us to solve for variables such as time (\( t \)). In our exercise, the difference \( \log y - \log a \) helps us find out how many times does our base have to be multiplied to grow from \( a \) to \( y \) — a key concept in financial growth over time.

Compound Interest Formula

Understanding the Formula

The compound interest formula is at the heart of calculating investment growth over time. It tells us how much an investment will be worth after interest has been added during multiple periods. The basic formula for compound interest is \( A = P(1 + r/n)^{nt} \), where:

• \(A\) is the amount of money accumulated after \( n \) years, including interest.
• \(P\) is the principal amount (the initial sum of money).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times that interest is compounded per year.
• \(t\) is the time the money is invested for, in years.

By reworking this formula and using logarithms, as we did in the textbook exercise, we can solve for \( t \) when the other variables are known, enabling us to calculate how many years are needed for the investment to reach a specific amount.

Time Value of Money

The time value of money is a core concept in financial mathematics that reflects the idea that a specific amount of money has different values at different points in time. The foundational principle is that money available today is worth more than the same amount in the future due to its potential earning capacity.

This principle is why investors seek to understand the future value of their investments. By using the compound interest formula, they can calculate the future value of their current investment. Furthermore, the formula adapted in our exercise uses logarithmic equations to pinpoint the exact time frame needed for an investment to grow to its desired value, emphasizing the relationship between time and money in the realm of finance.

Financial Mathematics

Application in Real Life

Financial mathematics is an area of applied mathematics concerned with financial markets and operations. It involves tools and concepts like present and future value, interest rates, annuities, and more. These concepts help individuals and businesses in planning investments, loans, retirements, and other financial activities.

In our case, understanding financial mathematics allows us to appropriately apply the compound interest formula and interpret logarithmic equations within the context of finance. This understanding translates into practical skills that can be employed in personal finance decisions, such as determining the best time to invest or the proper moment to withdraw investments to maximize returns.